Method and apparatus for modulation of guided plasmons

ABSTRACT

The present invention relates to a method and apparatus for the generation, manipulation and detection of plasmon, and/or solitons, both linear and non linear, in semi-conductor heterostructures. The apparatus includes a galium arsonide substrate, a two-dimensional electron gas well (2-DEG) ( 160 ) form thereon and a thin layer of Aluminum galium arsonide (AlGaAs) placed thereover. Launcher ( 130 ) and receiver lines ( 132, 134, 136 ) are formed on the AlGaAs layer. Each of the launcher ( 130 ) and receiver lines ( 132, 134, 136 ) includes coplanar waveguides which preferably consist of two to three metal lines. One line is interconnected with a photoconductive switch ( 120 ) and can be pulsed. The other line or lines are grounded. Pulsing the line forms an electric field which can be detected. Between the launcher and ( 130 ) and the receiver ( 140 ), the plasmon can be modulated by one or more additional gates line extending over the 2-DEG.

This Application is a International 371 of PCT/US98/08765 filed Apr. 30,1998 which claims benefit of Provisional No. 60/045,320 filed May 1,1997.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to the generation, manipulationand detection of plasmons in semi-conductor heterostructures, and moreparticularly to solitons and plasmons that can be utilized as a mediumfor signal transport, amplification and processing at high frequencies.

2. Related Art

In the past, surface acoustic wave (SAW) technology has been used togenerate surface acoustic waves in the megahertz frequency range. Thistechnology includes a launcher comprising a plurality of inter-digitatedteeth and a receiver, positioned some distance away from the launcher,which likewise contains a plurality of inter-digitated teeth. Byapplying a signal to the inter-digitated teeth, a compression or surfacewave is launched on the material and the wave moves throughout thematerial. The wave is a function of the frequency of the voltage appliedbetween the inter-digitated teeth. The voltage applied between theinter-digitated teeth launches the surface wave on the material whichmoves across the material and the wave travels to the receiving set ofinter-digitated teeth where it develops a voltage across that set ofinter-digitated teeth. When more than two sets of inter-digitated teethare used, one can be used to excite a surface wave which travels to thesecond set of inter-digitated teeth which modulates the signal whichmodulated signal can then be received by the third set ofinter-digitated teeth.

What is needed, and has not heretofore been developed, is a method andapparatus for increasing the frequency of a surface wave from themegahertz range to the terahertz range while reducing the scale of thesurface wave to a single electron layer. The applications of such adevice would include a source of Terahertz radiation for Terahertzfrequency range spectroscopy, signal processing and sensing.

Much attention has been focused on the collective excitation spectrum oflow-dimensional semiconductor systems. Interest has been driven both bythe search for novel behavior from bulk semiconductor systems, and bythe challenge to fabricate and probe devices based on theselow-dimensional structures. Collective excitations have been studiedextensively both experimentally and theoretically, using linearizedhydrodynamic models and quantum many-body formulations. The results fromthese models yield similar behavior in the ω,k regime, where k<k_(F),the Fermi wave vector. Thusfar in the literature, there has been notreatment of the non-linear behavior of these collective excitations inlow-dimensional systems. Furthermore, the non-linear regime has not beenexperimentally accessible due to the weak coupling schemes used forexcitation, including prism and grating couplers.

Previous efforts in this area are as follows:

U.S. Pat. No. 5,612,233 to Rosner, et al., discloses a method ofmanufacturing a single electron component in silicon MOS technique withat least two gate levels. The first gate level is made up of fine layershaving dimensions <100 nanometers with the surface and side walls of thegate level provided with an insulating layer. The second gate levelcovers the fine components of the first gate level in at least theregion of the active zone. The location and dimensions of nodes orpotential barriers of the single electron component are defined by thefine structures of the first gate level.

U.S. Pat. No. 5,485,018 to Ogawa, et al., discloses a logic device onthe nanometer scale providing multiple logic levels made up ofasymmetrically coupled quantum point contacts and a coupling regionbetween gate electrodes. The quantum mechanical carrier wave functionwithin the region is a spatially asymmetric wave. By changing the energylevel, the conductance of the device can be switched between multiplestable conductance levels. The device can be utilized to provide amulti-level output switched in response to terahertz pulses provided byan array of optical detectors.

U.S. Pat. No. 5,291,034 to Allam et al., discloses a nonlinear opticaldevice which employs a GaAs, AlGaAs well configuration which laterallyconfines a two-dimensional electron gas to produce laterally asymmetricquantum dot structures which can be controlled by bias potentialsapplied to alter the lateral extent of the well. This controls theassymetry of the well and affects the non-linear optical characteristicin response to incident radiation.

U.S. Pat. No. 5,239,517, to Mariani, discloses a coplanar wave guidewherein the ground planes taper directly into the outer bus bars of theSAW transducer and the center conductor of the coplanar waveguide feedsdirectly to the “hot” center electrode structure of the SAW transducer.

U.S. Pat. No. 5,105,232 to del Alamo, et al., discloses a quantum fieldeffect directional coupler. The device is comprised of two quantumwaveguides closely spaced apart. An adjacent gate electrode is providedover the space between the waveguides. The probability density betweenthe wave guides is controlled by a voltage applied to the gateelectrode, which controls coherent electron tunneling between waveguides. The design allows for several couplers to be connected in orderto perform multi-tasking operations.

U.S. Pat. No. 5,067,788 to Jannson, et al., discloses a high speedsurface plasmon wave modulator. The modulator employs a polymer glassmaterial and super fast electro-optic controlling medium for modulatinglaser light at ultra high frequencies. The modulator may be configuredin a planar format by complying a guided light wave with a surfaceplasmon wave generated at the interface of a metal electrode and anelectro-optic material.

U.S. Pat. No. 4,783,427 to Reed, et al., discloses a process for thefabrication of quantum well devices, where the quantum-wells areconfigured as small islands of GaAs in AlGaAs matrices. The dimensionsof such well devices are on the nanometer scale. These wells arefabricated by growing an n− on n+ epitaxial GaAs structure, which isthen etched to an e-beam defined pattern twice, and AlGaAs isepitaxially regrown each time.

U.S. Pat. No. 4,581,621 to Reed discloses quantum-coupled devices wherethere are at least two closely situated well devices. The bias betweenthe two closely situated wells can be adjusted in order to align theenergy levels of the two wells so that tunneling will occur veryrapidly, on the angstrom scale. Whereas conversely, when energy levelsare not aligned, tunneling will be greatly reduced. The inventionfurther provides for the output from these quantum well devices to becoupled to macroscopic contacts.

U.S. Pat. No. 4,205,329 to Dingle, et al., discloses a technique for thesynthesis of single crystal super lattices of semiconductor alloys,particularly with the fabrication of periodic structures of GaAs andAlAs. The patent further describes possible uses of the invention forwaveguides, heterostructure lasers and x-ray reflectors.

None of these efforts, taken either alone or in combination, teach orsuggest all of the elements of the present invention, nor disclose allof the benefits thereof.

OBJECTS AND SUMMARY OF THE INVENTION

It is a primary object of the present invention to provide a method andapparatus for the generation, manipulation, and detection of plasmons insemi-conductor heterostructures.

It is another object of the present invention to launch an electron waveto create a signal which can be read by a receiver.

It is an additional object of the present invention to provide a plasmawave that can be modulated between the launching and receiving thereof.

It is an additional object of the present invention to provide a devicethat can allow for the propagation of a plasmon by creating a structurecomprising a two-dimensional electron gas well formed between theinterface of a high bandgap semiconductor such as AlGaAs, and a lowerbandgap semiconductor such as GaAs.

It is an additional object of the present invention to provide a mediumfor the transport amplification and processing of signals at highfrequencies.

It is still an additional object of the present invention to provide amedium for transport amplification and processing of signals in theterahertz frequency range.

It is an additional object of the present invention to provide a devicewhere a plasmon is created by pulsing a current through a conductorplaced in close proximity to a two-dimensional electron gas well.

It is an additional object of the present invention to provide areceiver for a plasmon signal wherein the plasmon wave induces anelectric signal in the receiver.

It is an additional object of the present invention to utilize aphotoconductive switch to create a pulse in a conductor to induce aplasmon.

It is an additional object of the present invention wherein thephotoconductive switch can be illuminated by a laser on a pico orsub-pico second basis to create a pulse to create a plasmon.

It is an additional object of the present invention to modulate aplasmon wave it travels in a two-dimensional electron gas well.

The present invention relates to a method and apparatus for thegeneration, manipulation and detection of plasmons, and/or solitons,both linear and non-linear, in semi-conductor heterostructures. Theapparatus includes a heterostructure such as a Galium Arsonide (GaAs)substrate, a two-dimensional electron gas (2-DEG) well formed thereonand a thin layer of Aluminum Galium Arsonide (AlGaAs) placed thereover.Alternatively, GaAs could be used as a substrate and a higher bandgapmaterial such as AlGa as could be used as the thin layer. Launcher andreceiver lines are formed on the AlGaAs layer. Each of the launcher andreceiver lines includes a coplanar transmission line such as coplanarwave guides which preferably consist of two to three metal lines. Oneline is interconnected with a photoconductive switch and can be pulsedThe other line or lines are grounded. Pulsing the line forms an electricfield that creates an electron wave or plasmon in the 2-DEG. This wavepropagates to the receiver which includes two or more lines in which theplasmons induce an electric field which can be detected. Between thelauncher and the receiver, the plasmon can be modulated by one or moreadditional gate lines extending over the 2-DEG.

BRIEF DESCRIPTION OF THE DRAWINGS

Other important objects and features of the invention will be apparentfrom the following Detailed Description of the Invention taken inconnection with that accompanying drawings in which:

FIG. 1 is a top schematic view of an apparatus of the present inventionfor generating and detecting plasmons in semi-conductorheterostructures.

FIG. 2 is a top schematic view of another embodiment of the apparatusshown in FIG. 1.

FIG. 3 is a perspective view of the photoconductive switch shown in FIG.1.

FIG. 4 is a perspective view of the device shown in FIG. 3 which showsthe electric field created when the photoconductive switch is pulsed.

FIGS. 5 a and 5 b show the wave forms that can be created by the devicesof FIG. 1 and FIG. 2 respectively.

FIG. 6 is a perspective view of another embodiment of the device shownin FIG. 1 which device includes modulating gates positioned between thelauncher and receiver.

FIG. 7 is a cross-sectional view depicting the electron densitymodulation under a pair of biased gates.

FIG. 8 is a graph of the evolution of the Gaussian pulse along thez-axis.

FIG. 9 is a graph of a succession of frames two Gaussian pulses ofdifferent amplitude and pulse width propagating towards one another.

DETAILED DESCRIPTION OF THE INVENTION

The present invention relates to a device technology based on thegeneration, manipulation, and detection of plasmons, and/or solitons,both in the linear and non-linear regime, in semiconductorheterostructures. A plasmon is a collective, longitudinal charge densitywave. A solution is a wave form that can propogate without dispersing,i.e. without changing shape over long distances. These solitons, as wellas the plasmons, can be exploited as a medium for signal transport,amplification, and processing.

Plasmons have frequencies, determined by the charge carrier density,which lie in the far-infrared (FIR) sub-millimeter range and can betuned electrically by applying a bias voltage to an overlaid gate. Thefrequencies of the plasmons disperse over a wide range with wave-vector.For a 2-D electron gas, the dispersion relation is given by:${\omega (k)} = {\frac{n_{s}^{2}}{2m^{*}ɛ_{0}ɛ_{r}}k}$

where ω is the plasmon frequency (in the absence of any spatialmodulation of the 2-DEG density), k is the wave-vector in the plane ofthe 2D system, n_(s) is the areal density of the carriers of charge eand effective mass m, and ε_(r) is the effective dielectric function forthe system, which depends on the thicknesses and the dielectricproperties around the 2D charge sheet. These 2-D plasmons have beenobserved in AlGaAs/GaAs heterostructures, where the electrons areconfined in a very narrow potential well at the interface between theAlGaAs and GaAs layers, forming the 2-DEG. Furthermore, 1-D plasmonshave also been observed, where additional confinement of the electronsis produced in a 2^(nd) direction by a Schottky gate.

Direct coupling between EM radiation and plasmons is not possiblebecause the plasmon wavelength is always longer than that of freelypropagating EM radiation of the same frequency. Therefore, a couplingstructure is needed such as a prism coupler or an overlaid metalgrating, such as that shown in FIG. 1. Referring to FIG. 1, a topschematic view of a chip generally indicated at 20, having a launcherand a receiving coupling structure, is shown. The launcher 30 andreceiver 40 comprise coplanar wave guides formed of overlaid metalgrating, each consisting of three lines 32, 34, and 36, and 42, 44, and46 respectively. Importantly, any appropriate transmitter, such as acoplanar transmission line can be used for the launcher and receiverstructure. Air bridges 50 are formed on ground lines 32 and 36 and 42and 46 to tie them to the same potential. Each line is approximately 0.1microns wide and located about 1 micron apart. Other coupling structuresarc considered within the scope of the invention. It should be notedthat one such suitable structure would comprise two coplanar gates.However, three gates provides for greater accuracy and maintains theintegrity of the signal. The launcher 30 launches the wave and thereceiver 40 receives the wave. The carriers of the wave are electrons.

The wave travels across the chip, generally indicated at 20 through awell 60 comprising a thin electron layer, e.g. a 2-dimensional electrongas (2-DEG). The well 60 is formed on a heterostructure such as GaliumArsonide (GaAs) substrate, with a thin layer of Aluminum Galium Arsonide(AlGaAs), about 500 angstroms, placed over the well. This can be seen inFIG. 7, which shows a GaAs substrate 22 has a layer of AlGaAs layer 24thereon forming a well 60 therebetween wherein the 2-DEG resides. Otherheterostructures such as GaAs based, as well as non-GaAs basedheterostructures can be used. For example, a GaAs/InGaAs heterostructurecan be used. Importantly, the well 60 is positioned between theinterface of a high bandgap material and a low bandgap material. Eitherthe high or low bandgap material can serve as the substrate, the otherserving as the thin upper layer.

The launcher 30 comprising lines 32, 34, and 36 are on the upper layerand it stimulates the electron layer. When a signal is applied to thelauncher 30, as will hereinafter be described, it excites acompressional wave in the electron layer which modulates the electrondensity. So instead of a riple under surface like a SAW device, this isjust a change in the electron density. The receiver 40, comprising gatelines 42, 44, and 46, picks up and reads the signal. A third set oflines can be used to modulate the signal. Basically this scheme allowsfor what was done with SAW technology, but at much higher frequencies,such as in the terahertz (Thz) range.

A switch, such as a photoconductive switch 70 is used to create a pulsein line 34 to excite the 2-DEG. Referring to FIG. 3, an enlargedperspective view of such a photoconductive switch 70 is shown. Thephotoconductive switch 70 is essentially an optoelectronic device, thatcan be illuminated by a laser on a pico or sub-pico second basis. Thedevice includes Ohmic contacts 72. The center conductor 34 is biased tosome potential with respect to the two outer ground conductors 32 and 36positioned on either side of the center conductor 34. As a result of thelaser 74 illuminating the switch 70, the switch 70 is closed over a fewpico seconds and an electromagnetic transient or pulse propagating downthe line 34 is created. The pulse excites the electrons in the 2-DEG anda plasmon or wave, generally indicated at 62 in FIG. 1, is created. Theplasmon 62 travels from the launcher 30 to the receiver 40 where thesignal can be received. Pads 54 at the ends of the lines 34 and 44 areused to interface the chip 20 to the outside world. The signals are sofast that, they have to be sampled to determine the voltage which isoutput to the pad 54.

Referring to FIG. 4, input bars 32 and 34 are shown positioned on GaAs.When electric line 34 is pulsed, a longitudinal electric field iscreated based on electric field lines extending from the pulse line 34to the ground line 32. The strong electric field charge that is createdoperates in the well 60 to excite the electrons and create a plasmawave. The electrons are free to propagate along the well 60, and do sountil they reach the receiver 40, which is positioned about 20 to 50microns away from the launcher 30.

The receiver 40 operates in the same way as the launcher 30 in that thereceiver detects electromagnetic pull caused by the plasma waves whichbiases the conducting lines of the receiver 40 as the wave 62 passunderneath the receiver 40. The signal can then be processed by adetection circuit or an oscilloscope or some other device.

Referring now to FIG. 2, an alternate embodiment of the device shown inFIG. 1 is shown schematically. In this Figure, like reference numeralsrefer to like elements. Basically, the device generally indicated at 120includes a 2-DEG well 160. The launcher 130 includes lines 132, 134, and136, lines 132 and 136 being ground lines. Line 134 is interconnectedwith a photoconductive switch (not shown). Likewise, the receivergenerally indicated at 140, includes lines 142, 144, and 146, lines 142and 146 being ground and line 144 interconnected with a photoconductingswitch (not shown). Fingers 180 extend away from the lines 132, 134, 136and 142, 144 and 146 perpendicularly therefrom. It is these fingers 180that create the pulsed electric field. By applying various schemes tothe transducer geometry, the impulse response of the transducer and thetype of wave that can be generated can be varied as shown in FIGS. 5 aand 5 b. FIG. 6 b shows an apodization scheme imposed on the wavecreated by the overlap between the fingers 180 which allows forvariation of the frequency in bandwidth of the surface wave so that muchmore complex wave forms carrying more information can be generated.

Referring to FIG. 6, it can be seen that a plasma wave 62 is created bya launcher 30 which creates a plasmon in the 2-DEG which propagatestowards the receiver 40. This plasmon can be modulated by modulatingstrip lines or gates 90 to created a modulated plasmon 92 which isreceived by the receiver 40. The modulating gates 90 modulate theaverage electron density. As shown in FIG. 7, the gates 90 arepositioned over upper layer 24 which is placed over substrate 22 forminga 2-DEG well 60 therebetween. This spatial modulation leads tomodifications in the resonances. According to how many gates are appliedand the biasing scheme, the plasmon frequency is shifted higher or lowerthan the frequency in the absence of spatial modulation. These shifts inplasmon frequency can be measured and observed and can be exploited fordevice application. Typical sizes of metallic fingers are about 0.5 to0.7 microns with separations about 0.2-0.4 microns. The frequencyresponse of the system depends upon the electron density. Therefore, byapplying modulating gates 90 with biases along the propagation path theproperties of the gas can be altered which alters the property of theplasmon.

The metal fingers 180 of the grating spatially modulate the EM field ofthe normally incident radiation, generating spatial harmonics of thelongitudinal (z) component of the electric field of wave-vectork_(z)=2πn/d=nG in the plane of the 2-DEG. Here, d is the period of thegrating, G is the fundamental grating wave-vector, and n takes bothpositive and negative integer values. The grating period is much smallerthan the wavelength of the radiation and consequently only the zerothspatial harmonic of the field, E_(z) ^(nG)=E_(z) ⁰ propagates and allhigher orders are evanescent. The plasmon oscillation of spatialfrequency mG and temporal frequency ω^(mG) is driven by the evanescentfield E_(z) ^(nG) with n=m.

The effective dielectric function ε_(r) used above includes thescreening effect of the metal grating and the other semiconductor layersadjacent to the 2-DEG. For a standard AlGaAs/GaAs heterostructure, ε_(r)is known to be:

 ε_(r)=½(ε_(GaAs)+ε_(AlGaAs) ^(tanh(kh)))

where h is the distance between the 2-DEG and the top surface.

The grating introduces a lateral modulation of the conductivity close tothe 2-DEG but is not an efficient coupling mechanism. Pulsed excitationof the 2-DEG is used to achieve strong coupling to plasmons. Strongcoupling occurs because of the strong lateral component of the electricfield propagating along the coplanar transmission line. The generationof high-amplitude plasma oscillations leads to the formation ofsolitons. These solitons are of special interest as a source of THzradiation and for device application.

The devices of the present invention allow for exploiting plasmons inthe linear regime for device application as well as for exploiting thenonlinear plasmons, or solitons, for applications in other areas.Solitons are natural carriers of binary information because (a) they canpropagate long distances with changing shape (in a 2-DEG long meansabout 20 microns) (b) they can interact with each other without changingtheir shape or velocity (unlike linear waves), and (c) they exhibit athreshold energy, below which they disperse and above which they becomeself-contained. So again, by biasing metal fingers along the propagationpath, the average density is modulated and hence the spectralcomposition of the solution is modulated. When the average density islow enough (for sufficient positive bias to the gates), a solution willnot be able to propagate. This is a source of “1”'s and “0”'s in thesystem. Thus, the devices of the present invention have potentialapplication as digital logic gates. The nature of solution propagationprovides a natural thresholding operation—a pulse can only propagate asa solution if it contains at least the solution critical energy (whichdepends on the average electron density), otherwise it rapidlydisperses. This digital nature of solution propagation makes solitonsnatural carriers of binary information.

With respect to the replication of SAW technology using plasma waves inplace of Rayleigh waves, for SAW's and plasmons, the frequency launchedis the same as the frequency of the applied EM signal. However, thewavelengths for the SAW's and plasmons are much smaller than for theapplied EM signal. For example, for f=1 GHz,$\lambda_{EM} = {\frac{3 \times 10^{10}{{cm}/s}}{10^{9}{1/s}} = {30\quad {cm}}}$$\lambda_{plasmon} = {\frac{10^{7}\quad {{cm}/s}}{10^{9}{1/s}} = {{0.1\quad {cm}} = {100\quad {microns}}}}$$\lambda_{Acoustic} = {\frac{10^{5}\quad {{cm}/s}}{10^{9}{1/s}} = {{{.0001}\quad {cm}} = {1\quad {micron}}}}$

So when an applied EM signal reaches a SAW based or plasmon baseddevice, the “brakes are applied” and the signal is slowed down. This isuseful for delay line purposes. Similarly, the grating coupler andgrating detector can be employed in the same way as the SAW transducer.The width of each finger of the transducer and the separation betweenthe fingers determine the amount of the delay.

Similarly, the filtering capabilities of the SAW devices can bereproduced in the sub-millimeter range exploiting the plasmons. Thefinger-pair overlap of the inter-digitated fingers determines thefiltering characteristics. By applying a transducer-like structure as aload on a co-planar transmission line, the filtering and delaycharacteristics of SAW devices can be reproduced exploiting theplasma-waves instead of the Rayleigh waves. The frequency componentsneeded to couple to the plasma waves can be produced via aphoto-conducting switch fabricated on the co-planar transmission linessuch that when a pico or sub-pico second pulse produced by the lasercloses the switch, an EM pulse of corresponding width (containingfrequency components extending into the THz regime) is launched andpropagates along the co-planar transmission line toward the transducerload.

Surface wave filters are implemented via the repeated delaying andsampling on an input signal and are thereby classified as transversalfilters. The ID (interdigitated) transducer response is limited to abandwidth that is determined by the range of electrode spacings.Moreover, the electrode-pairs function as taps which sample thepropagating plasmon. By apodizing the transducer electrodes, meaningvarying the overlap length of the electrodes, the filter response can beadjusted. Furthermore, the device can be exploited as plasma waveconvolvors/correlators that operate with the same principles as SAWconvolvers/correlators.

Mathematical Model

A new class of localized excitations to form part of the non-linear modespectrum of the 2-DEG have been found. These are stable, pulse-likewaveforms that propagate with constant velocity and profile. Inaddition, these solitary excitations exhibit the properties of solitonsin that they retain their shape and velocity in a collision. For typical2-DEG electron densities on the order of 10¹² cm⁻², the characteristicfrequency of oscillation of the electron gas, ω_(p), extends into theTHz regime; consequently, these solitons are of special interest as asource of THz radiation and for device application.

These excitations are shown to be described by a nonlinear equation ofevolution in space which is suitable for the analysis of experiments inwhich the generation and detection of the excitations occur at twodifferent spatial locations. For this purpose, a matrix formulation ofthe model system of equations is used, which permits the extraction ofthe desired equations of evolution.

For the system under consideration, the interface in which electronmotion is free will be considered as the y-z plane, while the x-axiswill denote the direction perpendicular to this plane for which electronmotion is confined. For simplicity, it is assumed that the wave functionis exactly confined in the plane of the interface (y-z plane), with noextension outside of the plane; therefore, this model provides a gooddescription of non-linear excitations at low temperature, where only onesubband is occupied.

The dynamics of the electron gas are described via a hydrodynamic model,with the electrons possessing an isotropic mass and the ions assumingthe form of a compensating uniform background. The variables thatcomprise the hydrodynamic model are: the electron density n(r,t) and thecurrent density J(r,t). The electron density consists of contributionsfrom a time-independent background n₀ and time-dependent n,(r,t); i.e.,n(r,t)=n₀+n₁(r,t) where n₁(r,t)<n₀. For charges confined to a singlelayer (the y-z plane), these variables can be expressed asn(r,t)=n(z,t)δ(x) and J(r,t)=J_(z)(z,t)δ(x){tilde over (z)}, where z isa unit vector in the plane of the 2-DEG and, for simplicity, variationsin the y-direction are neglected.

These variables satisfy the continuity equation and the Euler equationof motion, namely:

∂_(t)(−en(r,t))+Δ·J(r,t)=0  (1)

$\begin{matrix}\begin{matrix}{{{\partial_{t}{J\left( {r,t} \right)}} - {\frac{1}{{en}_{0}}{\nabla{\cdot \left( {{J\left( {r,t} \right)}{J\left( {r,t} \right)}} \right)}}}} = \quad {{\frac{^{2}}{m}{n\left( {r,t} \right)}{E\left( {r,t} \right)}} +}} \\{\quad {{e\quad \beta^{2}{\nabla\quad {n\left( {r,t} \right)}}} - {\frac{1}{\tau}{J\left( {r,t} \right)}}}}\end{matrix} & (2)\end{matrix}$

Retardation effects are included in that the field variables, E(r,t) andB(r,t) obey the full Maxwell's equations. β is the spatial dispersionparameter with units of speed, and τ is the momentum relaxation time;both parameters are assumed to be velocity independent. As a result,this formulation does not include microscopic phenomena (such asexchange, correlation effects, and wave-particle interactions); however,it sufficiently describes the non-linear plasmon-polariton excitationsin the k<k_(F) regime. Furthermore, the formulation presented below canbe extended to include a more complete description of the electron gas.

The non-linear effects arising from the Δ·(J(r,t)J(r,t)) andn(r,t)E(r,t) terms in equation (2) are due to the fact that elements ofthe 2-DEG with different average current tend to compress, and the forcedriving a change in the average current density of the 2-DEG elementdepends on the local density, respectively. Thusfar in the literaturewhere the hydrodynamic model has been applied to 1-D and 2-D systems,eqn. (2) is linearized for the purpose of obtaining the linear plasmonspectrum.

For our purposes, Maxwell's equations are expressed through the vectorpotential A(r,t) in the gauge where the scalar potential vanishes. Inthis gauge, the vector potential is related to the electric and magneticfields such that E(r,t)=−∂_(t)A(r,t) and B(r,t)=Δ×A(r,t), and satisfiesthe wave equation $\begin{matrix}{{\left\lbrack {{\nabla^{2}{- \mu_{0}}}ɛ\partial_{t}^{2}} \right\rbrack {A\left( {r,t} \right)}} = {{{- \mu_{0}}{J\left( {r,t} \right)}\overset{\sim}{z}} - {\frac{e}{i\quad \omega \quad ɛ}{\partial_{z}{n\left( {r,t} \right)}}\overset{\sim}{z}}}} & (3)\end{matrix}$

The system of equations (1)-(3) forms the basis for the investigationsof the non-linear excitations.

The approach for deriving the equation of evolution in space for theseexcitations is as follows: (a) cast the model equations into a matrixformulation; (b) transform the coupled system of equations into thefrequency domain in order to isolate the plasmon excitations; and (c)inverse transform back into the time domain. This approach permits theanalysis of non-linear excitations which initially contain a widespectrum of frequencies (such as pulse excitations).

To this end, the gradient operator is decomposed as Δ=Δ_(t)+∂_(z){tildeover (z)}, where Δ_(t) is the part of the gradient operator transverseto the propagation direction (the z-direction) of the excitations. Thesystem of equations (1)-(3) is then transformed into the frequencydomain and cast in matrix form as a non-linear equation of evolutionalong the z-direction:

Γ∂_(z) ²ψ(r,ω)+L(Δ_(t),ω)ψ(r,ω)=NL[ψ(r,ω)]  (4)

where the linear operator L(Δ_(t),ω) consists of a lossless, Hermitianoperator L₀=L_(τ=∞) and a smaller, perturbative term L_(coll) (such thatL=L₀+L_(coll) and L₀>>L_(coll)):${{L_{0}\left( {\nabla_{t}{,\omega}} \right)} = \begin{bmatrix}{{\frac{\omega_{p}}{c}\left\{ {{\left( {{\partial_{x}^{2}{+ \omega^{2}}}\mu_{0}ɛ} \right)\overset{\sim}{z}\overset{\sim}{z}} - {{\partial_{x}\overset{\sim}{z}}\overset{\sim}{z}}} \right\}} + {\omega^{2}\mu_{0}ɛ\quad \overset{\sim}{x}\overset{\sim}{x}}} & {\mu_{0}{\delta (x)}\overset{\sim}{z}\overset{\sim}{z}} \\{\mu_{0}{\delta (x)}\overset{\sim}{z}\overset{\sim}{z}} & {{\delta (x)}{\left( \frac{\mu_{0}{mc}}{e^{2}n_{0}} \right)^{2}\left\lbrack {{\overset{\sim}{z}\overset{\sim}{z}} - {\frac{^{2}\beta^{2}}{\kappa_{z}^{2\quad}}\overset{\sim}{x}\overset{\sim}{x}}} \right\rbrack}}\end{bmatrix}}$ $L_{coll} = {{\begin{bmatrix}0 & 0 \\0 & {{\delta (x)}\frac{\mu_{0}{mc}}{^{2}n_{0}\omega_{p}}\frac{i}{\tau \quad \omega}\overset{\sim}{z}\overset{\sim}{z}}\end{bmatrix}\quad {\psi \left( {r,\omega} \right)}} = \begin{bmatrix}{{{A_{z}\left( {x,z,\omega} \right)}\overset{\sim}{z}} + {{\partial_{z}{A_{x}\left( {x,z,\omega} \right)}}\overset{\sim}{x}}} \\{\frac{^{2}n_{0}}{\mu_{0}{mc}^{2}}\left( {{{J_{z}\left( {z,\omega} \right)}\overset{\sim}{z}} + {i{\partial_{z}{n\left( {z,\omega} \right)}}\overset{\sim}{x}}} \right)}\end{bmatrix}}$ $\Gamma = {\begin{bmatrix}{- \left( {{{\partial_{x}\overset{\sim}{x}}\overset{\sim}{z}} - {\overset{\sim}{x}\overset{\sim}{x}}} \right)} & 0 \\0 & {{\delta (x)}c\frac{\mu_{0}m\quad \beta^{2}}{^{2}n_{0}}\left( {{\frac{1}{e\quad \omega}\overset{\sim}{z}\overset{\sim}{z}} + {\frac{1}{\kappa_{z}^{2}}\overset{\sim}{x}\overset{\sim}{z}}} \right)}\end{bmatrix}\quad {and}}$${NL} = {{- \frac{\mu_{0}m}{^{2}n_{0}\omega}}{\delta (x)}{\overset{\sim}{z}\begin{bmatrix}0 \\{\int{\int{{\omega_{1}}{\omega_{2}}{\delta \left( {\omega - \omega_{1} - \omega_{2}} \right)}\left\{ {{\frac{2}{{en}_{0}}J_{z\quad \omega_{1}}} - {\frac{e}{m}\frac{\omega_{1}}{\omega_{2}}A_{z\quad \omega_{1}}}} \right\} {\partial_{z}J_{z\quad \omega_{2}}}}}}\end{bmatrix}}}$

To obtain this form, the vector potential and the wave equation (3) havebeen written in component form. To simplify the analysis, the dielectricconstant, ε, will be used to characterize both the AlGaAs and GaAssurrounding the 2-DEG.

Solutions to (4) are sought in the form of an expansion in terms of theeigenfunctions of the operator, L₀, namely $\begin{matrix}{{\Psi \left( {r,\omega} \right)} = {\sum\limits_{\alpha}{{\Phi_{\alpha}\left( {x,\omega} \right)}{a_{\alpha}\left( {z,\omega} \right)}}}} & (5)\end{matrix}$

where Φ_(α)(x,ω) are the eigenfunctions of L₀ defined as

L₀(−iω,∂ _(x))Φ_(α)(x,ω)=κ_(zα) ²(ω)ΓΦ_(α)(x,ω)  (6)

The eigenfunctions Φ_(α)(x,ω) possess the orthonormality property(Φ_(α)(x,ω),ΓΦ_(β)(x,ω))=2δ_(αβ) in the x-direction. Equation (6) issolved by obtaining expressions for the regions x>0 and x<0 andsatisfying boundary conditions. The two boundary conditions are foundfrom the continuity of the tangential component of the vector potentialand from the integration of equation (6) about x=0. The solutions havethe form: $\begin{matrix}{{\Phi_{\alpha}\left( {x,w} \right)} = \begin{bmatrix}{\frac{{- i}\quad \omega \quad m}{^{2}n_{0}}\left( {1 - \frac{\beta^{2}\kappa_{z\quad \alpha}^{2}}{\omega^{2}}} \right){^{{- {\kappa_{z\quad \alpha}}}x}\left( {\overset{\sim}{z} \mp {\frac{\kappa_{z\quad \alpha}^{2}}{\kappa_{x}}\overset{\sim}{x}}} \right)}} \\{\left( {i\quad \omega} \right)\left( {\overset{\sim}{z} - {\frac{\kappa_{z\quad \alpha}^{2}}{e\quad \omega}\overset{\sim}{x}}} \right)}\end{bmatrix}} & (7)\end{matrix}$

where c is the speed of light in the medium;${\kappa_{x\quad \alpha}} = \sqrt{\kappa_{z\quad \alpha}^{2} - {\omega^{2}/c^{2}}}$

is the transverse wave number, and the top (bottom) sign is forsolutions x>0 (x<0).

The boundary conditions yield the well known dispersion relation forlinear 2-D plasmons, namely, $\begin{matrix}{\omega^{2} = {{\beta^{2}\kappa_{z\quad \alpha}^{2}} + {\omega_{p}c\sqrt{\kappa_{z\quad \alpha}^{2} - \frac{\omega^{2}}{c^{2}}}}}} & (8)\end{matrix}$

where $\omega_{p} = \frac{n_{0}^{2}}{2\quad ɛ\quad {mc}}$

is the 2-D plasma frequency. At very low frequencies, ω<<ω_(p), thetransverse wave number |κ_(x)|→0 and the dispersion relation tends toω≈cκ_(z); hence, the mode structure contained in equation (7) becomeselectromagnetic-like in that the longitudinal component of the wavevector A_(z)→0. At the other extreme, where${\omega\operatorname{>>}\frac{\omega_{p}}{\beta/c}},$

the excitations also approach a non-dispersive wave, traveling at thethermal speed with dispersion relation ω≈βκ_(z). Plasmon behaviordominates in this limit, characterized by the transverse component ofthe wave vector A_(x)→0. In the region between these two extremes, theexcitations are mixed and dispersive, due to the non-negligiblecontributions of both the transverse and longitudinal field componentsto the mode structure.

At low temperatures and for a 2-DEG of infinitesimal thickness, theelectrons occupy one sub-band. In this case, there exists only one modeof propagation, i.e. α takes on only one value so thatΦ_(α)(x,ω)≡Φ(x,ω).

Employing the eigenfunctions, equation (7), their orthogonalityproperty, and the expansion in eqn. (5), the equation for the modalamplitude a_(α)(z,ω)≡a(z,ω) in the frequency domain follows from eqn.(4): $\begin{matrix}\begin{matrix}{{\partial_{z}^{2}{a\left( {z,\omega} \right)}} = \quad {{{- {\kappa_{z}^{2}(\omega)}}{a\left( {z,\omega} \right)}} - {\frac{i\quad \omega}{2\quad \tau \quad c^{2}}{a\left( {z,\omega} \right)}} +}} \\{\quad {\frac{1}{2\quad {en}_{0}c^{2}}{\int{\int{{\omega_{1}}{\omega_{2}}{\delta\left( {\omega - \omega_{1} -} \right.}}}}}} \\{\left. \quad \omega_{2} \right)\quad \left( {\omega_{1}\omega_{2}} \right)\left\{ {2 + {\frac{\omega_{1}}{\omega_{2}}\left( {1 - \frac{\beta^{2}{\kappa_{z}^{2}\left( \omega_{1} \right)}}{\omega_{1}^{2}}} \right)}} \right\} {a\left( \omega_{1} \right)}{\partial_{z}{a\left( \omega_{2} \right)}}}\end{matrix} & (9)\end{matrix}$

Attention is focused on non-linear pulse excitations in theplasmon-polariton regime with bandwidth ω<(ω_(p)c/β), where retardationis significant and the effects of spatial dispersion are weak. In thisregime, the dispersion relation, equation (8), can be approximated as$\kappa_{z}^{2} \approx {\frac{\omega^{2}}{c^{2}}{\left( {1 + {\frac{\omega^{2}}{\omega_{p}^{2}}\left( {1 - \frac{\beta^{2}}{c^{2}}} \right)^{2}}} \right).}}$

Using this approximation in equation (9) and inverse transforming intime yields a non-linear equation in space-time of the form:$\begin{matrix}\begin{matrix}{{\partial_{z}^{2}{a\left( {z,t} \right)}} = \quad {{\partial_{t}^{2}{a\left( {z,t} \right)}} - {K^{2}{\partial_{t}^{4}{a\left( {z,t} \right)}}} + {\frac{1}{2{\tau\omega}_{p}}{\partial_{t}{a\left( {z,t} \right)}}} +}} \\{\quad \left\{ {{\frac{1}{2}{\partial_{t}^{2}{\partial_{z}\left( {a^{2}\left( {z,t} \right)} \right)}}} - {{a\left( {z,t} \right)}{\partial_{t}^{2}{\partial_{z}{a\left( {z,t} \right)}}}} +} \right.} \\\left. \quad {{K^{2}{\partial_{t}^{2}{a\left( {z,t} \right)}}{\partial_{z}{a\left( {z,t} \right)}}} + {K^{2}\frac{\beta^{2}}{c^{2}}{\partial_{t}^{4}{a\left( {z,t} \right)}}{\partial_{z}{a\left( {z,t} \right)}}}} \right\}\end{matrix} & (10)\end{matrix}$

where ${K = {1 - \frac{\beta^{2}}{c^{2}}}};$

t and z have been normalized such that t→ω_(p)t and$\left. z\rightarrow{z\frac{\omega_{p}}{c}} \right.;$

and the excitation amplitude a(z,t) has been normalized such that$\left. {a\left( {z,t} \right)}\rightarrow{\frac{\omega_{p}}{2{en}_{0}c}{{a\left( {z,t} \right)}.}} \right.$

This is a non-linear, dispersive equation which is second order inspace, implying solutions that propagate in the +z and −z directions. Insearch of the nature of the solution, the damping term on the right handside of equation (10) in the limit τ→∞ is considered and later the scaleof the dynamics with τ is compared.

Using the reductive perturbation method, the non-linear evolutionequation (10) can be shown to belong to a class of nonlinear evolutionequations which admits a reduction to the well-known Kortweg-de-Vries(KdV) equation. Introducing a small parameter, ε<<1, and expressing theamplitude a(z,t) in a perturbation series,${a = {\sum\limits_{n = 0}{a_{n}ɛ^{n}}}},$

the following change of variables is introduced:

Z=ε³ z

T=ε^(½)(z−t)

Substituting these relations into equation (10) and integrating oncewith respect to time, we find to lowest order in ε:

∂_(Z) {circumflex over (a)}(Z,T)+K²∂_(T) ³ {circumflex over(a)}(Z,T)−(1+K/2){circumflex over (a)}(Z,T)∂_(T) {circumflex over(a)}(Z,T)e ^(−Z/ω) ^(_(p)) ^(τ)=0  (11)

where a₁(Z,T)={circumflex over (a)}(Z,T)e^(−Z/ω) ^(_(p)) ^(τ). In thelimit τ→∞, equation (11) reduces to the KdV equation, which describesthe evolution of nonlinear disturbances of small amplitude in plasmasand other types of dispersive media exhibiting dispersion of the typeκ=aω(1±b²ω²).

The evolution of the non-linear excitations from an initial arbitrarywaveform with frequency bandwidth ω<(ω_(p)c/β) is obtained by numericalsolution of eqn. (10). A Gaussian temporal profile is used (withamplitude and width denoted by a₀ and w₀, respectively) as theexcitation at the point z=0, and the response of the 2DEG at variouslocations z>0 has been calculated. This corresponds to a feasibleexperimental configuration where the plasmon generation, via pulsedexcitation, and detection occur at two different locations in space.

From numerical solution, we find that eqn. (10) admits a new class oflocalized excitations. FIG. 8 demonstrates the evolution of the Gaussianpulse into two such excitations at various values of the propagationdirection, z. Note from FIG. 8 that the excitations are propagating inpositive and negative time, such that only one pulse is observed fortimes t>0 at a particular location z>0. The shape of the excitation isunchanged for larger values of z.

FIG. 9 represents the results of the computer calculation for thecollision of two of these localized, non-linear excitations. Equation(10) is solved with initial amplitude a(z,t) consisting of two Gaussianpulses of different amplitude and pulse width, shown at z=0, propagatingtowards one another. FIG. 9 depicts a succession of frames in space ofthe collision process. This figure demonstrates that these non-linearexcitations do not scatter upon collision. Each initial Gaussian pulseevolves into two pulses, propagating in opposite directions. Thecolliding pulses emerge from the collision retaining the same velocityand profile with which they entered and thus behave as solitons, herebyreferred to as 2-DEG solitons.

Note from eqns. (5) and (7) that the current density in the 2-DEGassociated with the excitation is given by J_(z)(x,z,ω)=φ^(J) ^(_(z))(x,ω)a(z,ω) for the single mode, where φ^(J) ^(_(z)) (x,ω) is thecontribution to the eigenfunctions from the current density. Using theexpression for the φ^(J) ^(_(z)) (x,ω) given in equation (7) andinverting back to the time domain, we find thatJ_(z)(x,z,t)∝∂_(t)a(z,t). The time derivative of the modal amplitude isthus directly proportional to the current density, a physical,measurable quantity.

The strength of the damping term in equation (10) is of the order(2ω_(p)τ)⁻¹≈0.003, evaluated in the Drude model for the high densitiesand mobilities (n=2.0×10¹² cm⁻²;μ=2.0×10⁶ cm²/Vs) that are presentlybeing achieved in GaAs/AlGaAs heterostructures. Including this term inthe numerical solution of equation (10) (using an initial Gaussian pulsewith parameters a₀=0.4, w₀=10.0), the 2-DEG solution amplitude decays by10% within a normalized distance z≈120. As can be seen from FIG. 9, thisdistance is much greater than the solution evolution length, z≈60.Therefore, neglect of the damping term is justified in this model withina propagation distance of twice the solution evolution length.

In conclusion, an equation of evolution for the non-linear excitation ofa 2-DEG with a frequency bandwidth ω<(ω_(p)c/β) has been presented.Numerical solution of this equation illustrates that an arbitrary inputwaveform will evolve in space into solitons containing a broad frequencyspectrum that extends into the terahertz regime. As the search forbetter coupling schemes to plasmons continues, these results demonstratea promising potential for devices for terahertz frequency applications.

Having thus described the invention in detail, it is to be understoodthat the foregoing description is not intended to limit the spirit andscope thereof. What is desired to be protected by Letters Patent is setforth in the appended claims.

What is claimed is:
 1. An apparatus for generating, manipulating anddetecting plasmons in semi-conductor heterostructures comprising: asubstrate; upper layer of material placed over the substrate; atwo-dimensional electron gas well formed between the upper layer ofmaterial and the substrate; launching means applied on the upper layerover the well; and receiving means positioned on the layer over thewell, the launching means launching a plasmon that travels through thewell to the receiving means where it can be received.
 2. The apparatusof claim 1 further comprising modulating means positioned on the upperlayer between the launching means and the receiving means for modulatingthe signal.
 3. The apparatus of claim 1 wherein the launching meanscomprises first and second coplanar wave guides.
 4. The apparatus ofclaim 3 wherein further including means for providing a pulsedelectrical current to the first coplanar waveguide.
 5. The apparatus ofclaim 4 wherein the first coplanar waveguide is interconnected with aphotoconductive switch.
 6. The apparatus of claim 5 further comprisinglaser means for operating and closing the photoconductive switch.
 7. Theapparatus of claim 1 wherein the launching means comprises threecoplanar waveguides, the central waveguide being conductive and theoutside waveguides being grounded.
 8. The apparatus of claim 7 whereinthe receiver comprises three coplanar waveguides.
 9. The apparatus ofclaim 3 wherein one of the substrate or the upper layer comprises a highbandgap material and the other of the substrate or the upper layercomprises a lower bandgap material.
 10. The apparatus of claim 9 whereinone of the substrate or the upper layer comprises AlGaAs and the otherof the substrate or the upper layer comprises GaAs.
 11. A method ofusing a plasmon in a semi-conductor heterostructure to allow forterahertz signaling comprising the steps of: forming a two-dimensionalelectron gas well between a substrate and an upper layer, positioninglaunching means on the upper layer above the well; positioning receivermeans on the upper layer over well; launching a plasmon at the launchingmeans; and receiving the plasmon at the receiving means.
 12. The methodof claim 11 wherein the launching means comprises coplanar waveguidesand the step of launching a plasmon comprises sending a pulsed electriccharge through the coplanar waveguides to excite the electrons in thewell therebelow.
 13. The method of claim 12 wherein the receiving meanscomprises coplanar waveguides and the step of receiving the plasmoncomprises propagating the plasmon through the well below the receivingcoplanar waveguides and inducing a charge in the receiving coplanarwaveguides which can be detected.
 14. The method of claim 13 furthercomprising the step of modulating the plasmon as it travels between thelaunching means and the receiving means by modulating means positionedon the upper layer over the well.
 15. The method of claim 14 wherein themodulating means comprises a gate and the step of modulating the plasmoncomprises charging the gate to change the spatial arrangement of atwo-dimensional electron gas in the well.